Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{3x - 24}{x^2 - x - 72} \times \dfrac{6x^2 - 54x}{x - 8} $
Answer: First factor the quadratic. $p = \dfrac{3x - 24}{(x - 9)(x + 8)} \times \dfrac{6x^2 - 54x}{x - 8} $ Then factor out any other terms. $p = \dfrac{3(x - 8)}{(x - 9)(x + 8)} \times \dfrac{6x(x - 9)}{x - 8} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ 3(x - 8) \times 6x(x - 9) } { (x - 9)(x + 8) \times (x - 8) } $ $p = \dfrac{ 18x(x - 8)(x - 9)}{ (x - 9)(x + 8)(x - 8)} $ Notice that $(x - 8)$ and $(x - 9)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ 18x(x - 8)\cancel{(x - 9)}}{ \cancel{(x - 9)}(x + 8)(x - 8)} $ We are dividing by $x - 9$ , so $x - 9 \neq 0$ Therefore, $x \neq 9$ $p = \dfrac{ 18x\cancel{(x - 8)}\cancel{(x - 9)}}{ \cancel{(x - 9)}(x + 8)\cancel{(x - 8)}} $ We are dividing by $x - 8$ , so $x - 8 \neq 0$ Therefore, $x \neq 8$ $p = \dfrac{18x}{x + 8} ; \space x \neq 9 ; \space x \neq 8 $